{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Maroon Emphasis" -1 258 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Purple Emphasis" -1 260 "Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 261 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 262 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Blue emphasis" -1 263 "Times" 0 0 0 0 255 1 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" 258 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 258 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 262 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "Grey Em phasis" -1 272 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{PSTYLE "N ormal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 } {PSTYLE "" 3 259 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 3 "" 0 "" {TEXT 259 20 "The Remez algorithm:" }}{PARA 259 "" 0 "" {TEXT 271 56 " .. constructing minimax polynomial approxi mations for " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT 269 5 " and " }{XPPEDIT 18 0 "cos*x" "6#*&%$cosG\"\"\"%\"xGF%" }{TEXT 270 4 " on " }{XPPEDIT 18 0 "[-Pi/4,Pi/4]" "6#7$,$*&%#PiG\"\"\"\"\"%! \"\"F)*&F&F'F(F)" }{TEXT 268 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 37 "by \+ Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 20 "Ver sion: 25.3.2007\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 " " 0 "" {TEXT -1 68 "load interpolation and function approximation proc edures including: " }{TEXT 0 5 "remez" }}{PARA 0 "" 0 "" {TEXT -1 17 " The Maple m-file " }{TEXT 272 10 "fcnapprx.m" }{TEXT -1 37 " contains \+ the code for the procedure " }{TEXT 0 5 "remez" }{TEXT -1 25 " used in this worksheet. " }}{PARA 0 "" 0 "" {TEXT -1 123 "It can be read into a Maple session by a command similar to the one that follows, where t he file path gives its location. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "read \"K:\\\\Maple/procdrs/fcnapprx.m\";" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT 0 5 "remez" } {TEXT -1 10 ": examples" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 "; " }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT 266 8 "Question" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 22 "(a) Find a polynomial " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" } {TEXT -1 29 " of degree 14 to approximate " }{XPPEDIT 18 0 "cos*x" "6# *&%$cosG\"\"\"%\"xGF%" }{TEXT -1 17 " on the interval " }{XPPEDIT 18 0 "[-Pi/4,Pi/4]" "6#7$,$*&%#PiG\"\"\"\"\"%!\"\"F)*&F&F'F(F)" }{TEXT -1 121 " with minimax relative error, giving the coefficients to a pre cision of 20 digits.\n(b) Plot the relative error curve for " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 4 ". " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 267 8 "Solution" }{TEXT -1 2 ": " }}{PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 137 "evalf[20](remez(cos(x),x=-Pi/4..Pi/4,14,\n 'ma xerr','minerr',type=even,errtype=relative,info=true));\np := unapply(% ,x);\nmaxerr;\nminerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~alg orithm:~calculating~minimax~error~estimate~by~solving~a~rational~equat ionG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D--~minimising~the~RELATIVE~e rror~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\qstandard~Chebyshev~points~for~initial~critical~poin ts~that~lie~in~the~right~half~of~the~interval:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6+$\"\"!F$$\"=@-+\"[m\"R331eBK:!#G$\"=A\"e/d`8t@%\\')e0IF '$\"=l<_,H!)z)=lSQMO%F'$\"=Q7&)p(3y&zpsOg`bF'$\"=RO:DL@u?BkqMIlF'$\"=( o>U9Nvd[.)G8csF'$\"=\"fVb,ea[f\"z&pIq(F'$\"=g%3m:'4$[uRj\")R&yF'" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%,iteration~1G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%G----------------- ---------------------G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Fprovisiona l~polynomial~approximation:G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#,2$\"@ R1^CUk*******************!#J\"\"\"*&$\"@w&*4Q\\vG***************\\F&F' )%\"xG\"\"#F'!\"\"*&$\"@mO_`kV*3VmmmmmmT!#KF')F,\"\"%F'F'*&$\"@c%QFc%* Q\"3!f)))))))))Q\"!#LF')F,\"\"'F'F.*&$\"@FD'f$*p0#=j\"HG(e,[#!#NF')F, \"\")F'F'*&$\"@ZHSgXSTP.poGJdv#!#PF')F,\"#5F'F.*&$\"@EZXIaNn.Mi*Rbb(3# !#RF')F,\"#7F'F'*&$\"@iy9USjyLRW+68_8\"!#TF')F,\"#9F'F." }}{PARA 11 " " 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Rk\"R&=l9!#H" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%>goal~for~rela tive~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"=8?ArWk#oj)=`0 88!#L" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%Uerror~graph~drawn~for~the~right~half~of~the~intervalG " }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "67-%'CURVESG6% 7[v7$$\"\"!F)$\"=+++++++++!*[vdN!#Z7$$\"=)=$G_'=O5Io`q?9#!#I$\"=zw`N98 ^t)G[%\\aNF,7$$\"=vjc/tB2-mt59%G%F0$\"=n<8_-$QE\"4S\">Za$F,7$$\"=j&\\o &f&3J!\\5;@EkF0$\"=8.n(\\t2]?pVY%GNF,7$$\"=]F84YZ9/KZ@Go&)F0$\"='=i)=l r-UVYcq0NF,7$$\"=7*p8>r@1)4ABC&G\"!#H$\"=,B%y0[,JM02&*4W$F,7$$\"=]l#=# \\*G3k%Hkl8D1)[4z ;J/%FE$\"=ni[axiut.E\\%oX#F,7$$\"=k\"Qh^G\"oz-cQ`\")[FE$\"=1IOkvJ$)4hN -I#*>F,7$$\"==^9K<7k3l$\\3bs&FE$\"=dS%H)Rjar%\\'ppn9F,7$$\"=t?:[\\6gPF JJ[plFE$\"=+fsw)oG^*G&*Ro1!*!#[7$$\"=V8]c3yUnw/lW4uFE$\"=w(yOI'\\q;S#Q \")45$Feo7$$\"=91&[wYasf#y)4%\\#)FE$!=!\\lBe'f0NJgi6+HFeo7$$\"=h!))>,S %)z$G[M;G!*FE$!=#QCU!zAXT!**zB7S)Feo7$$\"=3b7fKVryI=q\"p!)*FE$!=\"*zQG %e`lZjF\"Rq8F,7$$\"=QlajLig+HBrKh5!#G$!=20J\"zfiL&>5Yi%)=F,7$$\"=D0=,M 5M$\\Zai>9\"F]q$!=Z(G^5M%\\L\"y]\"*)\\BF,7$$\"=SfK*f%>xE(H\"eND7F]q$!= 6BEbxLMKfGIDmFF,7$$\"=b8Z(z&G?g>\"3\\(38F]q$!=!Q]@\"=g*eVr$*)p/JF,7$$ \"=9s0`kdW0:1\\(=R\"F]q$!=;\\Hsj\\ohBaN\\aLF,7$$\"=uIk3r')o]5J2+v9F]q$ !=y=L%[!G?(zPE?#4NF,7$$\"=a]d.#o4<7K,xj\\\"F]q$!=!=YD'G$Q2JAsnH`$F,7$$ \"=Nq])HpIF>`H`x^\"F]q$!=L\"pC;(o$RM2AC+b$F,7$$\"=;!RMRq^PEudH\"R:F]q$ !=;i%Rh*)oDUi7L,MV[)=k]NF,7$$\"=Qp;tZd$yaeqMYi\"F]q$!=i?@k5TRt.P&HQ`$F,7$$\" =>*)4oen&)='z)4,Y;F]q$!=`dO2vyAjfZ!\\-^$F,7$$\"=rZ')z)*[RIi^RK@%G:pj'z\"F]q$!=a_cSx%>f#G8-dcJF,7$ $\"=96'[s/SsRXZ@9)=F]q$!=o/1y&o,\"*Q/8g5#GF,7$$\"=1;4ebqa_zLg?m>F]q$!= P\\!\\(4?^]yz'3$)R#F,7$$\"=Y?$>K)[@qfE(Q80#F]q$!=`0#R(f!*fnh-&\\#)*=F, 7$$\"='[sd3r#)y)R>9ZO@F]q$!=lXGleO7#\\\\q:sL\"F,7$$\"=<:H7IAiwhVD^=AF] q$!=mr>iK_kt#z))zja(Feo7$$\"=Z0\")Q\\&[O,T\"y]x*pZ \"Feo7$$\"=gp,Lr1OuIo[0vBF]q$\"=_,)y,Q(eG=:')3)4%Feo7$$\"=tLAF$ffLy(og b\\CF]q$\"=]!)Rz@nBUG>()e$f*Feo7$$\"=)**42C6?i[??X\"QDF]q$\"=))*oJ9jL3 J0LwOe\"F,7$$\"=Dm>aJ13*=`LMni#F]q$\"=.(R))*3(3OF>]f_:#F,7$$\"=H!y@0-? ea9&4y,FF]q$\"=Qf6v6@xeMMKh#e#F,7$$\"=M%f,&4%fD!fnv#ox#F]q$\"='HOCQA]q m]x+b%HF,7$$\"=HQNE@_e!=HiCT'GF]q$\"=p:#=ec6;jAuBJF$F,7$$\"=C#[DI.6'eC y;U^HF]q$\"=I5>]y^iqI^^x'[$F,7$$\"=:Gkd`=,o=@!e+*HF]q$\"=4*=j)G#y=:eel Ba$F,7$$\"=1ut7uETx7kVpGIF]q$\"=aG/xqPb$e\\)e\"Q-*[,y!Hz:zNF,7$$\"=(*>$yY\\8ooqqIt1$F]q$\"=<,WV`6@o+o%4) yNF,7$$\"=#Hz`\\!R^\"R&y)[m3$F]q$\"=%RT2l50X+!3x5sNF,7$$\"=*eEH_J9i4+0 nf5$F]q$\"=?/RF\\>Km%f\"3/fNF,7$$\"=LCGA.bAR[Luu!>$F]q$\"=Mv(HE&He\"eH X_mU$F,7$$\"=y#Q;7pOAep\"y_vKF]q$\"=k6(Q`^nvg/0&\\uJF,7$$\"=rZ+p.x#**R 3zeiN$F]q$\"=vX%G.:sG(o#pc\"HGF,7$$\"=j7P;;(=w@Zw*)pV$F]q$\"=)zA_bL%4d W!Q#e!R#F,7$$\"=t3qqoC&3Kn1C7_$F]q$\"=8=M%f>DV^Q+/s%=F,7$$\"=#[I]7A'3C uo$eag$F]q$\"=.V4O.<)['z(QePB\"F,7$$\"=pLT:s;=64`<\"Go$F]q$\"=t(*[=h&[ ,I5AT]E'Feo7$$\"=ciz0BrF)Ru8l,w$F]q$!=iJWXi\"fuK.gwwI%!#]7$$\"=sjC'f\" ec\"RD@,O%QF]q$!=_DSp,==$[Mx_!*)oFeo7$$\"=(['p')3X&[QwGPq#RF]q$!=#z=!p 9&*)*z2)4PKN\"F,7$$\"=*)eK_*3!z\\Q?Rq8SF]q$!=(z')o5Qlw-wK9Y*>F,7$$\"=! 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)F,\"\"%F'F'*&$\"5Q/'*e)))))))))Q\"!#AF')F,\"\"'F'F.*&$\"54YY**GG(e,[# !#CF')F,\"\")F'F'*&$\"5/dU'plGJdv#!#EF')F,\"#5F'F.*&$\"5FP;;d9bb(3#!#G F')F,\"#7F'F'*&$\"5L'4wh2K7_8\"!#IF')F,\"#9F'F." }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,2$\"5'******* ************!#?\"\"\"*&$\"5%G***************\\F/F0)9$\"\"#F0!\"\"*&$\" 5d-VmmmmmmT!#@F0)F5\"\"%F0F0*&$\"5Q/'*e)))))))))Q\"!#AF0)F5\"\"'F0F7*& $\"54YY**GG(e,[#!#CF0)F5\"\")F0F0*&$\"5/dU'plGJdv#!#EF0)F5\"#5F0F7*&$ \"5FP;;d9bb(3#!#GF0)F5\"#7F0F0*&$\"5L'4wh2K7_8\"!#IF0)F5\"#9F0F7F(F(F( " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"59b#4$R-D3%e$!#R" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#$\"59BA@XG\"zSe$!#R" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "plot(p(x),x=-Pi/4..Pi/4);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6%-%'CURVESG6$7S7$$!3_***p#=;)R&y!# =$\"3'GAqH#y1rqF*7$$!3B0)e>s#f6vF*$\"3'>1g_s\")*3tF*7$$!3\"\\\">-u>o8s F*$\"3$Rg0NdG!4vF*7$$!3--WX!=]'yoF*$\"32!)3O;D.ExF*7$$!3jY?oyvRTlF*$\" 3d[%)[!f;d$zF*7$$!3'>qb'zyu0iF*$\"3UY+=-NWN\")F*7$$!3[(31DkdX*eF*$\"3Z jB%eZLCJ)F*7$$!3#)f'znT Z*F*7$$!3UK#>%3PwfHF*$\"3S`:%R;y^c*F*7$$!3q'[w&y6w0EF*$\"3E)*zn!4_.d>F*$\"3\"oz&fx564)*F* 7$$!3k66.fKD[;F*$\"3%\\aa#4.Zk)*F*7$$!30oCc30Z48F*$\"3\\d$)Q*y'Q9**F*7 $$!3IQou\\@po)*!#>$\"31B)Hd'RM^**F*7$$!3Uk(Q@3'o-lF^r$\"3WN5(e)\\')y** F*7$$!31B2qe!)Qp\"pU***F*7$$\"3]V],SoR+kF^r$\"3!**3J 9XC&z**F*7$$\"3=eAT\\_Qc'*F^r$\"3woq#*GLT`**F*7$$\"3-o9]()R,-8F*$\"3kt 1^awN:**F*7$$\"3+$\\rGv\"4J;F*$\"3%4;!)>krs')*F*7$$\"3]G/t69\\\\>F*$\" 3KfW>I_d5)*F*7$$\"3zEvBZ*>II#F*$\"3ljf(f-vft*F*7$$\"3:yu?g5o?EF*$\"3!p ()=-Ui&e'*F*7$$\"3O`)4<5_)fHF*$\"3i7;b\"Q_^c*F*7$$\"3Kg'RgZ\">nKF*$\"3 7l\\vtQ+r%*F*7$$\"3Ke)[6f%>.OF*$\"3SR-4(QUyN*F*7$$\"3qX:8PtM>RF*$\"38W Q:fwrT#*F*7$$\"3Wlk&>0=)\\UF*$\"3,z=**[BY5\"*F*7$$\"3qA]C!e8Hd%F*$\"3U \")*=[07(o!=))F*7$$\"35RJ#HkSpB&F* $\"3Eex0bvxf')F*7$$\"3#QM0Ik$4qbF*$\"37**[rT!*R)[)F*7$$\"3'*[:/oz[+fF* $\"39<,r%HN\"4$)F*7$$\"3ZIEKRg3/iF*$\"3z\"RjZv4k8)F*7$$\"3Y>b.!pV?b'F* $\"3F\"pMF=M#HzF*7$$\"39&Rm3;hK'oF*$\"3&[n$R&o$zNxF*7$$\"3s)\\%3'*Q3&> (F*$\"3yJ6O')zH@vF*7$$\"359_J$f!o7vF*$\"3C(fmWAR#3tF*7$$\"3_***p#=;)R& yF*F+-%'COLOURG6&%$RGBG$\"#5!\"\"$\"\"!F`[lF_[l-%+AXESLABELSG6$Q\"x6\" Q!Fe[l-%%VIEWG6$;$!+N;)R&y!#5$\"+N;)R&yF]\\l%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Thus " }{XPPEDIT 18 0 " p(x)=.99999999999999999996-.49999999999999999284*x^2+.4166666666666643 0257e-1*x^4-.13888888888858960438e-2*x^6+.24801587282899464609e-4*x^8- .27557312865696425704e-6*x^10+.20875555145716163727e-8*x^12-.113521232 07617609633e-10*x^14" "6#/-%\"pG6#%\"xG,2-%&FloatG6$\"5'************** *****!#?\"\"\"*&-F*6$\"5%G***************\\F-F.*$F'\"\"#F.!\"\"*&-F*6$ \"5d-VmmmmmmT!#@F.*$F'\"\"%F.F.*&-F*6$\"5Q/'*e)))))))))Q\"!#AF.*$F'\" \"'F.F5*&-F*6$\"54YY**GG(e,[#!#CF.*$F'\"\")F.F.*&-F*6$\"5/dU'plGJdv#!# 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(c) The relative error curve plotted over the interval " }{XPPEDIT 18 0 "[-Pi/4, Pi/4];" "6#7$,$*&%#PiG\" \"\"\"\"%!\"\"F)*&F&F'F(F)" }{TEXT -1 10 " is . . . 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Give the coefficie nts to a precision of 20 digits. " }}{PARA 0 "" 0 "" {TEXT -1 69 "(b) \+ Use the answer to (a) to find a minimax polynomial approximation " } {XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 18 " of degree 13 for " }{XPPEDIT 18 0 "sin*x" "6#*&%$sinG\"\"\"%\"xGF%" }{TEXT -1 17 " on \+ the interval " }{XPPEDIT 18 0 "[-Pi/4,Pi/4]" "6#7$,$*&%#PiG\"\"\"\"\"% !\"\"F)*&F&F'F(F)" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 50 "Giv e the coefficients to a precision of 20 digits." }}{PARA 0 "" 0 "" {TEXT -1 39 "\n(c) Plot the relative error curve for " }{XPPEDIT 18 0 "p(x)" "6#-%\"pG6#%\"xG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 8 "Solution" }{TEXT -1 2 ": " }} {PARA 0 "" 0 "" {TEXT -1 4 "(a) " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 139 "evalf[20](remez(sin(x)/x,x= -Pi/4..Pi/4,12,\n 'maxerr','minerr',type=even,errtype=relative,info =true)):\nq := unapply(%,x);\nmaxerr;\nminerr;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%]pRemez~algorithm:~calculating~minimax~error~estimate~ by~solving~a~rational~equationG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%D- -~minimising~the~RELATIVE~error~--G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #%!G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#%\\qstandard~Chebyshev~points~ for~initial~critical~points~that~lie~in~the~right~half~of~the~interval :G" }}{PARA 12 "" 1 "" {XPPMATH 20 "6*$\"\"!F$$\"o#))oFXxo*[F'$\"\\p;/,\\ShF'$\"0m&)>Q)GTH>wqF'$\"A`rZhmmmmmm;F&F')%\"xG\"\"#F'!\" \"*&$\"?PF$y2N3[+?LLLLL)!#KF')F,\"\"%F'F'*&$\"?%=,D\"yv3C.%G)p7%)>!#LF ')F,\"\"'F'F.*&$\"?ycs:69&[bB*H8tbF!#NF')F,\"\")F'F'*&$\"?rk]#3Omgoa l~for~relative~difference:G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"8'4A&3LF17$$\"<=$G_'=O5Io`q?9#F5$\"<>:$p?oQ51E;[2LF17$$\"KWku%)G$F17$$\"r@1)4ABC&G\"!#G$\"D1)[4z;J/%FY$\"<_XOOv(3SLP'zs\\#F17$$\"<:Qh^G\"oz-cQ `\")[FY$\"<6v_P(*4Wyx0Z![@F17$$\"<5X@t@T'3l$\\3bs&FY$\":Vic$F ^q7$$\"<.))>,S%)z$G[M;G!*FY$!<2TT*[KSj;_Pe<5F^q7$$\"9\"Far$!xE(H\"eND7Fa r$!<**ohX5N9r=&3[))=F17$$\"\"3\\(38Far$!V`NHwk0EF17$$\"<2V'3r')o]5J2+v9Far$!j[&3\")GF17$$\"<*4P)[rsZL&fe]g:Far$!</Uc[=eJYbAzKF17$$\"%G:pj'z\"Far$!\"\\>H!R=Far$!<[lGEs:>-xNYhF$F17$$\"<6h[s/SsRXZ @9)=Far$!Far$!<%p'Gcru&>ebyfjJF1 7$$\"Far$!FbNbyIF17$$\"K)[@qfE(Q80#Far$! <9&>:Gv3G!\\zu>&GF17$$\"<[sd3r#)y)R>9ZO@Far$!,^b#F17$$\"< ^\"H7IAiwhVD^=AFar$!aJ13*=`LMni#Far$\"D,eHhcF^q7$$\"F17$$\"$Far$\"#3\"**QI$*eJ$Far$\"8D@q!oT$Fa r$\" a5+lR^P3X^$>$F17$$\"@Ve')zZaB)3V/^#)*)p\"F^q7$$\"vGqwD^UFa r$!RP#*=P#)4Yhb$F^q7$$\"4h&R!yQFL%Far$!<&[mZ#R1wUEsD/<*F ^q7$$\"<$ocf:4-P]!*)>UT%Far$!<^%G\"38FR[hl*[b9F17$$\"<8Er#G[@x<]yR)\\% Far$!<0c))*4#>@N)e2_p>F17$$\"$R])zm[Far$!Sz 9!R7HEQ=n)[Far$!LN#=<0LF17$$\"N (3&\\Far$!0ij*>p\\[T3`5CF17$$\"<&yuWrizkOMiWF`Far$!DmaB*[&Far$!B]cFar$\"<,$pd8;'Q\")*Q3j-kF^q7$$\"<1:i1F6B!G/nJMdFar$\"<(e^+sy @$RK'[4F8F17$$\"F17$$\"<-50T[] '4'R:^D*eFar$\"^2z $)4'Far$\"<&[i_c'R%[*zDIhE$F17$$\"]SJ$F17$$\"<)*Ql5sIRN7tL#z hFar$\"<$y%>reZmj@#f(yI$F17$$\"<4zS)\\ZkVE&RZ%*>'Far$\">;'y(eL$Hf5m>iFar$\"3)Q4wAd&*)Q'Far$\"k*y@EJwg&>F17$$\"<\\%zz[$=W6V)QC6lFar$\"<$p1e_s%f7GY*f%f\"F17 $$\":p/uBp$f'Far $\"<'R,m8w7%[_:^lq(F^q7$$\"$F^q7 $$\"%[7R1xmFar$!<^`!>+[fizI\"zWX\"F^q7$$\"RCKGq\"o\\(=nFa r$!pzY3<'3V,oFar$!<%4]#)o)*H$pf7D<_\"F17$$\"<'o(>$=$eG6S]L([\\RC%>F17$$\"='Ro*eI/!HF17$$\"TpwG1OqFar$!V>`?=W+F$F17$$\"<@^1I QLPgs(>$y0(Far$!<3v+1P$3(3p.v-H$F17$$\"<+2PM)=+NaFlroqFar$!If^)e2gnGZD7rFar$!<;kyx-p9y-9WFH$F17$$\"<)f)*e&QW8f*y#R J7(Far$! 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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalf(expand(x*q(x)),20):\np := unapply(%,x);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%\"pGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,09$$\"5p '*****************!#?*&$\"5aZhmmmmmm;F0\"\"\")F-\"\"$F4!\"\"*&$\"5VW-+ KLLLL$)!#AF4)F-\"\"&F4F4*&$\"5C*H@SG)p7%)>!#BF4)F-\"\"(F4F7*&$\"5J94:! *H8tbF!#DF4)F-\"\"*F4F4*&$\"5Vz=%QYeq]]#!#FF4)F-\"#6F4F7*&$\"5&HmMCsj8 %*e\"!#HF4)F-\"#8F4F4F(F(F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 6 " Thus " }{XPPEDIT 18 0 "p(x) = .9999999999999999 9669*x-.16666666666666614754*x^3+.83333333333200024443e-2*x^5-.1984126 9828402129924e-3*x^7+.27557313299015091431e-5*x^9-.2505070584638418794 3e-7*x^11+.15894136372243466295e-9*x^13;" "6#/-%\"pG6#%\"xG,0*&-%&Floa tG6$\"5p'*****************!#?\"\"\"F'F/F/*&-F+6$\"5aZhmmmmmm;F.F/*$F' \"\"$F/!\"\"*&-F+6$\"5VW-+KLLLL$)!#AF/*$F'\"\"&F/F/*&-F+6$\"5C*H@SG)p7 %)>!#BF/*$F'\"\"(F/F6*&-F+6$\"5J94:!*H8tbF!#DF/*$F'\"\"*F/F/*&-F+6$\"5 Vz=%QYeq]]#!#FF/*$F'\"#6F/F6*&-F+6$\"5&HmMCsj8%*e\"!#HF/*$F'\"#8F/F/" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 55 "(c) The relative error curve plotted over the interval " }{XPPEDIT 18 0 "[-Pi/4, Pi/4];" "6#7$,$*&%#PiG\"\"\"\"\"%!\"\"F)*&F&F' F(F)" }{TEXT -1 10 " is . . . 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